Solving 3-Variable Systems of Equations

There are many ways to solve 3-variable systems of equations. The three most common are substitution, elimination, and Cramer's Rule. I chose to teach Cramer's rule because it builds off of previous concepts, and because it is the most straightforward (in my opinion) of the three methods.

To begin, you'll have a problem that looks something like this:
2x + y - z = 5
3x + 5y - 4z = 10
x - 2y + 2z = 12

where you will need to solve for x, y, and z, and your answer will be a coordinate triplet, meaning that it can be graphed on a 3-D coordinate plane. First, make a 3 x 3 matrix of all the coefficients to the left side of the equals sign. Find that determinant, and label it "D."

Next, make another 3 x 3 matrix but leave out the column of x's (so your first column should be blank). Where the x coefficients used to be, insert the numbers to the right of the equals sign. Find that determinant, label it "Dx." Repeat this process for both y and z, leaving out those variables' numbers and replacing them with the numbers to the right of the equals sign for each matrix. Find those determinants and label them "Dy" and "Dz."

Almost done! To get your final values of x, y, and z, simply divide the following to get your answer:
x = Dx/D,      y = Dy/D,      z = Dz/D